Karolin Stange,∗ Christoph Wick,† Playing Music in Just and Haye Hinrichsen∗∗ ∗Hochschule fur¨ Musik, Intonation: A Dynamically Hofstallstraße 6-8, 97070 Wurzburg,¨ Germany Adaptive Tuning Scheme †Fakultat¨ fur¨ Mathematik und Informatik ∗∗Fakultat¨ fur¨ Physik und Astronomie †∗∗Universitat¨ Wurzburg¨ Am Hubland, Campus Sud,¨ 97074 Wurzburg,¨ Germany Web: www.just-intonation.org [email protected] [email protected] [email protected] Abstract: We investigate a dynamically adaptive tuning scheme for microtonal tuning of musical instruments, allowing the performer to play music in just intonation in any key. Unlike other methods, which are based on a procedural analysis of the chordal structure, our tuning scheme continually solves a system of linear equations, rather than relying on sequences of conditional if-then clauses. In complex situations, where not all intervals of a chord can be tuned according to the frequency ratios of just intonation, the method automatically yields a tempered compromise. We outline the implementation of the algorithm in an open-source software project that we have provided to demonstrate the feasibility of the tuning method. The first attempts to mathematically characterize is particularly pronounced if m and n are small. musical intervals date back to Pythagoras, who Examples include the perfect octave (m:n = 2:1), noted that the consonance of two tones played on the perfect fifth (3:2), and the perfect fourth (4:3). a monochord can be related to simple fractions Larger values of mand n tend to correspond to more of the corresponding string lengths (for a general dissonant intervals. If a normally consonant interval introduction see, e.g., Geller 1997; White and White is sufficiently detuned from just intonation (i.e., the 2014). Physically, this phenomenon is caused by simple frequency ratio), the resulting mismatch of the circumstance that oscillators such as strings almost-coinciding partials leads to a superposition of emit not only their fundamental frequency but also waves with slightly different frequencies (Helmholtz a whole series of partials at integer multiples of 1877). The fast beating of these partials, which do the fundamental frequency. Consonance is related not quite coincide, can result in a sensation of to the matching of higher partials, i.e., two tones roughness, or of being out of tune, that ruins the with fundamental frequencies f and f tend to perception of consonance. be perceived as consonant if the m-th partial of With the historical development of fretted in- the first matches the n-th partial of the second. struments and keyboards, it made sense to define a In other words, mf = nf (see Figure 1). Although system of fixed frequencies in a pattern of repeating the perception of consonance is a highly complex octaves. The frequency ratios of stacked intervals psychoacoustic phenomenon (see, e.g., McDermott, multiply. (For example, the chord A2–E3–B3, con- Lehr, and Oxenham 2010; Stolzenburg 2015) that sisting of two perfect fifths each having the ratio also depends on the specific context (Parncutt and 3:2, yields a frequency ratio of 9:4 from A2 to B3.) Hair 2011; Milne, Laney, and Sharp 2015), one can This immediately confronts us with the fundamen- basically assume that the impression of consonance tal mathematical problem that multiplication and prime numbers are incommensurate in the sense Computer Music Journal, 42:3, pp. 47–62, Fall 2018 that powers of prime numbers never yield other doi:10.1162/COMJ a 00478 simple prime numbers. For example, it is impossible c 2018 Massachusetts Institute of Technology. to match k just perfect fifths with just octaves Stange et al. 47 Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/comj_a_00478 by guest on 27 September 2021 Figure 1. Consonance of a Helmholtz (1877), the fifth just perfect fifth. The figure is perceived as consonant shows a measured power because many partials of spectrum of the piano keys the corresponding natural A2 (110 Hz) and E3 harmonic series coincide (165 Hz). As first theorized (marked by the arrows in by Hermann von the figure). because (3/2)k = (2/1) for all k, ∈ N. Mathemati- keyboard). This means that music can be played in cally speaking, the concatenation of just musical any key, differing only in the global pitch but not in intervals (by multiplying their frequency ratios) is harmonic texture. an operation that does not close up on any finite set This high degree of symmetry can, however, of tones per octave. only be established at the expense of harmony Fortunately, the circle of fifths does approximate (cf. Duffin 2008). In fact, the only just interval a closure, with only a small mismatch: When in ET is the octave, with the frequency ratio 2:1, stacking twelve just fifths on top of each other, whereas all other intervals are characterized by the resulting frequency ratio (3/2)12:1 differs from irrational frequency ratios, deviating from the just that of seven octaves (ratio 27:1) by only a small intervals. For some intervals the variation is quite amount, explaining why the Western chromatic small and hardly audible. For example, the equally scale is based on twelve semitones per octave. The tempered fifth differs from the just perfect fifth remaining difference of approximately 1.4 percent by only two cents. For other intervals, however, (23.46 cents), known as the Pythagorean comma,is the deviations are clearly audible, possibly even nevertheless clearly audible and cannot be neglected disturbing. For example, the minor third in ET is in a scale with fixed frequencies. Likewise, a almost 16 cents smaller than the natural frequency sequence of four just perfect fifths transposed back ratio 6:5. The same applies to the major third, which down by two octaves ((3/2)4/22) differs from a major is about 14 cents greater than the ratio 5:4. These third of 5:4 by the so-called syntonic comma of discrepancies may explain why there was some 21.51 cents. reluctance among many musicians to accept ET. It Because it is impossible to construct a musical was not until the 19th century that ET became a scale that is based exclusively on pure beatless new tuning standard, presumably both because of intervals, one has to seek suitable compromises. Western music’s increasingly complex harmonies Over the centuries this has led to a fascinating and because of an increasing intonational tolerance variety of tuning systems, called temperaments, on the part of the audience. that reflect the harmonic texture of the music in the epochs in which they were developed (see, e.g., Barbour 2004). With the increasing demand of Just Intonation flexibility, equal temperament (ET) finally prevailed in the 19th century and has established itself Although musical temperaments provided a good as a standard temperament of Western music. solution for most purposes, music theorists and In ET, the octave is divided into twelve equally instrument makers have searched for centuries sized semitones with the constant frequency ratio for possible ways to overcome the shortcomings 1:21/12. The homogeneous geometric structure of ET of temperaments, aiming to play music solely on ensures that all interval sizes are invariant under the basis of pure intervals (see, e.g., Duffin 2006). transposition (i.e., horizontal displacement on the This is referred to as just intonation (JI). Tuning an 48 Computer Music Journal Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/comj_a_00478 by guest on 27 September 2021 Table 1. Five-Limit Tuning semitones 0 1 2 3 4 5 6 7 8 9 10 11 f : f ∗ 1:1 16:15 9:8 6:5 5:4 4:3 45:32 3:2 8:5 5:3 9:5 15:8 Five-limit tuning is the most common choice of frequency ratios in just intonation. The tuning gets its name from the fact that all terms in the ratios have prime factorizations using no primes larger than 5. instrument in just intonation means adjusting the where all tones are tuned statically in advance. twelve pitches of the octave such that all frequency In comparison, many other instruments (such as ratios are given by simple rational numbers with strings) allow the musician to recalibrate pitch respect to a certain reference frequency f ∗. For during performance, and the same applies, of course, example, a possible choice of such frequency ratios to the human voice. Musicians playing such in- is listed in Table 1. struments tune the pitches dynamically while the Just intonation always refers to the tonic of a music is being played. By listening to the harmonic given scale, referred to in this article as the keynote. consonance and its progression, well-trained musi- In its own reference scale, JI sounds very consonant, cians are able to estimate the appropriate frequency possibly even sterile, but a transposition to other intuitively and to correct their own pitch instanta- scales is, unfortunately, not possible. For example, neously. As pointed out by Duffin (2006), the notes tuning a piano in just intonation with keynote C, played are a compromise between just intervals a C-major triad sounds consonant, whereas most and the prevailing ET. By dynamically adapting the triads in other keys sound out of tune. The same pitches, performers can significantly improve the applies to modulations from one key to another. harmonic texture. It is in this context that we quote Thus, just intonation has the reputation of being the cellist Pablo Casals (taken from Applebaum and impractical, for good reason. Applebaum 1972), To overcome this problem, a possible solu- tion would be to increase the number of tones Don’t be scared if your intonation differs from per octave.